Subtracting Fractions Calculator

a/b − c/d = (a × (L ÷ b) − c × (L ÷ d)) ÷ L where L = lcm(b, d), the least common multiple of the two denominators Then reduce the difference

First numerator

The top of the fraction being subtracted from: 3 in 3/4. It is multiplied by however many times its own denominator goes into the common denominator — by 3, in the default case, which turns 3/4 into 9/12. It may be negative, and the sign travels with it.

First denominator

The bottom of the first fraction: 4 in 3/4. It must be positive, and it is one of the two numbers whose least common multiple becomes the common denominator. It stays in the panel as written, so the rewrite can be checked against it.

Second numerator

The top of the fraction being taken away: 1 in 1/6. Its multiplier is different from the first one's — 2 in the default case, giving 2/12 — and that asymmetry is exactly what the two converted readings show.

Second denominator

The bottom of the second fraction: 6 in 1/6. Together with the first denominator it fixes the common denominator at 12. It must be positive for the same reason the first one must be.

Common denominator (L)

The least common multiple of the two denominators: 12 for 4 and 6, and 77 for the coprime pair 7 and 11, where it does come out as the plain product. It is computed as one denominator divided by the greatest common divisor of the two, multiplied by the other, which is the same number by a route that does not overflow. It is printed on its own line because it is the number the whole method turns on, and because a common denominator larger than necessary is the most common arithmetic slip on this operation.

Converted fractions

The two fractions after both have been rewritten over the common denominator: 9/12 and 2/12 in the default case. They are printed as converted rather than reduced, since reducing them would hide the multiplier that did the work. The subtraction happens between these two numerators.

Difference

What is left after the two converted numerators are subtracted: 9 − 2 = 7 over 12. It is printed in lowest terms, so it can differ from what the subtraction of the converted numerators gave — 5/6 minus 1/3 leaves 3/6, which is printed as 1/2 — and it collapses to a bare 0 when the two fractions are equal.

The first use is the exercise itself, which is taught as soon as denominators differ, and the panel is arranged around the textbook procedure: find the common denominator, convert, subtract, reduce. The second is checking work done by hand, where the two converted readings are the useful part — if yours are not these, everything after them differs too. The third is a pair of denominators that share a factor, such as 4 and 6, where the common denominator is smaller than the product and the panel shows the saving. The fourth is a pair of coprime denominators such as 7 and 11, the other extreme: nothing is shared, the common denominator is the product, and the answer usually arrives already in lowest terms. The fifth is a pair of equal fractions, where the difference is zero and the panel prints it as a bare 0 — the one reading on this page that is not a fraction. It does not add, multiply or divide, and it does not take mixed numbers; those are three other pages.

1. 3/4 − 1/6

   1. Find the common denominator: the least common multiple of 4 and 6 is 12, not 24, because both divide 12 and 4 × 6 has counted their shared factor of 2 twice
   2. Convert the first fraction: 12 ÷ 4 = 3, so 3/4 becomes (3 × 3)/(4 × 3) = 9/12
   3. Convert the second fraction: 12 ÷ 6 = 2, so 1/6 becomes (1 × 2)/(6 × 2) = 2/12
   4. Subtract the numerators over the common denominator: 9 − 2 = 7, giving 7/12
   5. 7 and 12 share nothing, so 7/12 is already in lowest terms

   The default case, and the one that shows why the common denominator is not simply the product: 4 and 6 both divide 12, and 12 is the smallest number they both divide, so twelfths are enough and twenty-fourths would have been twice the work. The two converted readings are the evidence — 9/12 and 2/12 have the same denominators, which is exactly what makes the subtraction legal. Notice that the converted numbers are printed as they were converted and not reduced: 2/12 could be written as 1/6, but then the multiplier that produced it would disappear, and the multiplier is the step being shown.

2. 5/6 − 1/3

   1. Find the common denominator: 3 divides 6, so the least common multiple is 6
   2. The first fraction needs no converting: 6 ÷ 6 = 1, so it stays 5/6
   3. Convert the second: 6 ÷ 3 = 2, so 1/3 becomes 2/6
   4. Subtract: 5 − 2 = 3 over 6, giving 3/6
   5. Reduce: the greatest common divisor of 3 and 6 is 3, so 3/6 = 1/2

   The case that shows a common denominator does not finish the job. The difference of the converted numerators is 3/6, and 3/6 is not the answer — the answer is 1/2, because 3 and 6 still share a factor. A method that stops at the common denominator prints 3/6 here and is wrong in a way the page cannot see, which is why this row is in the table and in the worked examples rather than being left to chance. Note also that one of the two converted readings is unchanged: when one denominator divides the other, only one fraction has to be rewritten, and the reading that comes back as it went in is the honest record of that.

3. 1/7 − 1/11

   1. Find the common denominator: 7 and 11 share no factor, so the least common multiple is their product, 77
   2. Convert the first fraction: 77 ÷ 7 = 11, so 1/7 becomes 11/77
   3. Convert the second: 77 ÷ 11 = 7, so 1/11 becomes 7/77
   4. Subtract: 11 − 7 = 4 over 77, giving 4/77
   5. 4, 7 and 11 share no common factor, so 4/77 is already in lowest terms

   The other extreme of the common denominator: two denominators with nothing in common, so the least common multiple really is the product and both fractions have to be rewritten. This is the case where people who skip the least common multiple and always multiply the denominators get the same answer — and the reason it is worth showing anyway is that the previous example, 4 and 6, is the one they get wrong. Note that the answer is 4/77 and not something like 4/77 reduced: coprime denominators do not guarantee a reduced answer, they just make it likely, and the reduction is done on the result regardless.

4. 5/8 − 3/4

   1. Find the common denominator: 4 divides 8, so the least common multiple is 8
   2. The first fraction stays: 8 ÷ 8 = 1, giving 5/8
   3. Convert the second: 8 ÷ 4 = 2, so 3/4 becomes 6/8
   4. Subtract: 5 − 6 = -1 over 8, giving -1/8

   Both fractions are positive and the answer is not: subtracting a larger fraction from a smaller one takes the result past zero, and the panel says so with a minus sign rather than refusing the entry. The common denominator is 8 and the two converted readings are 5/8 and 6/8, which is what makes the outcome obvious before the subtraction is even done — the second numerator is the larger one. The sign lives on the numerator and nowhere else: the answer is -1/8, not 1/-8, and the denominator stays positive on every line of the panel.

5. 1/2 − 1/2

   1. The denominators are equal, so the least common multiple is that denominator, 2
   2. Neither fraction needs converting: both are already written over 2
   3. Subtract: 1 − 1 = 0 over 2, giving 0/2
   4. Reduce: the greatest common divisor of 0 and 2 is 2, so 0/2 becomes 0/1, which is written as a bare 0

   The case where the answer is not a fraction at all. Equal fractions have a difference of zero, and zero reduces to 0/1, which the fraction notation writes without a denominator — so the main reading is a bare 0 while the common denominator line still says 2 and both converted readings still say 1/2. That is the point: the two inputs were real fractions and the method really ran, and only the result collapsed. It is also the one place where the panel looks different from the same subtraction done on the adding page, where a sum only collapses when it reaches a whole number.

Both denominators must be positive, because a fraction carries its sign on the numerator — and the converted readings depend on that, since each numerator is multiplied by a whole number of times and the sign has to stay in one place. The numerators may be negative, and the difference may be negative even when both inputs are positive: 5/8 − 3/4 is -1/8, which is a correct answer and not an error. The numbers are limited in size. Each denominator is capped at 100000, and each numerator at about 4.5 × 10¹⁰, which is the largest value for which the two converted numerators still land on integers that can be represented exactly. Past that the subtraction would be quietly inexact, so the page refuses the entry instead of printing a number that looks right. Only whole numbers are accepted in any of the four boxes — a decimal numerator is rejected, and the page points at the decimal-to-fraction page instead. Zero is fine in a numerator: it converts to 0 over the common denominator like anything else. The page subtracts two fractions and nothing else: it does not add, multiply or divide, and it does not take a mixed number written as a whole number beside a fraction, which has to be turned into an improper fraction first — that is what the mixed number page is for. The intermediate readings here are deliberately different from the ones on the fraction calculator and the mixed number page: those two multiply the denominators straight across and print the product before reducing, while this page finds the least common multiple and prints the two converted fractions. The same subtraction therefore shows 9/12 and 2/12 here and a different intermediate pair there, and the answers agree. Finally, the answer is always reduced but the converted readings are not — a converted fraction can print as something like 6/8, and that is the reading recording the conversion rather than a missed reduction.

How do you subtract fractions with different denominators?

Rewrite both fractions over a common denominator, then subtract the numerators and keep that denominator. 3/4 minus 1/6 becomes 9/12 minus 2/12, which is 7/12. Multiplying the two denominators always gives a usable common denominator, and taking the least common multiple gives a smaller one.

Why is the common denominator 12 and not 24 for 3/4 and 1/6?

Because 12 is the smallest number that both 4 and 6 divide. Multiplying 4 by 6 counts the factor of 2 that the two denominators share twice, which doubles every number in the working. Twenty-fourths would reach the same answer with twice the arithmetic — the panel prints the smaller one so the multipliers stay small.

Do I still have to reduce after finding the common denominator?

Yes. A common denominator makes the subtraction possible, it does not make the result reduced. 5/6 minus 1/3 is 5/6 minus 2/6, which is 3/6, and 3/6 is 1/2. The page reduces the difference and leaves the two converted readings as they were converted, so both facts are visible.

Can the answer be negative?

Yes. 5/8 minus 3/4 is 5/8 minus 6/8, which is -1/8: subtracting a larger fraction from a smaller one takes the result below zero, and that is a real answer rather than an error. The sign sits on the numerator, so the answer is written -1/8 and never as a fraction with a negative denominator.

What happens when the two fractions are equal?

The difference is zero. 1/2 minus 1/2 converts to 1/2 minus 1/2, the numerators subtract to 0, and 0 over 2 reduces to 0 over 1 — which is written as a bare 0, without a denominator. The common denominator line still reads 2, because the method really ran.

Can I subtract mixed numbers here?

No. This page takes four whole numbers that make two fractions, so a mixed number such as 4 1/2 has to be turned into an improper fraction — 9/2 — before it can be entered, and that conversion is what the mixed number to improper fraction page does. Adding, multiplying and dividing are three other pages as well.

• Fraction — the quotient of two integers written with a numerator and a denominator, and the arithmetic of subtracting two of them — Wolfram MathWorld (United States)
• Least Common Multiple — the smallest positive integer divisible by two given integers, which is the common denominator this page converts both fractions to — Wolfram MathWorld (United States)
• Greatest Common Divisor — the largest integer dividing two given integers, used here both to find the least common multiple and to reduce the difference — Wolfram MathWorld (United States)
• Reduced Fraction — the lowest terms of a fraction, which is the form the difference is printed in and not the form the converted readings are printed in — Wolfram MathWorld (United States)
• Relatively Prime — two integers with no common factor greater than one, the case in which the common denominator is the plain product of the two denominators — Wolfram MathWorld (United States)